Graph polynomials for a class of DI-pathological graphs

Abstract

Let G = (V, E) be a simple graph. A dominating set D ⊆ V is a set such that the closed neighborhood of D is the entire vertex set. An independence set of a graph G is a subset of vertices that are pairwise non-adjacent. A DI-pathological graph is a graph where every minimum dominating set intersects every maximal independent set. Let d(G, i) denote the number of dominating sets of G of size i. The domination polynomial of a graph G is defined by D(G, x) = ∑ |V | i=γ (G) d(G, i)x i. Let s(G, i) denote the number of independent sets of size i in a graph G. The independence polynomial is defined by I (G, x) = ∑ α(G) i=0 s(G, i)x i. In this paper, we will examine the domination polynomial and the independence polynomial of a family of extremal DI-pathological graphs. We will further define an independent dominating set and examine the corresponding independent domination polynomial for these graphs.